climxrrelem - Mathbox for Glauco Siliprandi

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Theorem climxrrelem 41496

Description: If a seqence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxrrelem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxrrelem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climxrrelem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
climxrrelem.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climxrrelem.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
climxrrelem.p	⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
climxrrelem.n	⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))

**Assertion**
Ref	Expression
climxrrelem	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Distinct variable groups: 𝐴,𝑗 𝐷,𝑗 𝑗,𝐹 𝑗,𝑀 𝑗,𝑍 𝜑,𝑗

Proof of Theorem climxrrelem Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1874	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1874	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
3		nfra1 3171	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
4	2, 3	nfan 1863	. . . . 5 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
5	1, 4	nfan 1863	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
6		climxrrelem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	6	uztrn2 12082	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
8	7	adantll 702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
9		climxrrelem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
10	9	fdmd 6358	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑍)
11	10	ad2antrr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
12	8, 11	eleqtrrd 2871	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
13	12	adantlrr 709	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
14		simpll 755	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
15	8	adantlrr 709	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
16		rspa 3158	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
17	16	adantll 702	. . . . . . 7 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
18	17	adantll 702	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
19	9	ffvelrnda 6682	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
20	19	3adant3 1113	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ^*)
21		simpll 755	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝜑)
22		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) = -∞)
23		simpl 475	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) ∈ ℂ)
24	22, 23	eqeltrrd 2869	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
25	24	adantll 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
26		climxrrelem.n	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
27	21, 25, 26	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
28	27	adantlrr 709	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
29		fvoveq1 7005	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = -∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
30	29	adantl 474	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
31		simpl 475	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
32	30, 31	eqbrtrrd 4958	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
33	32	adantll 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
34	33	adantlrl 708	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
35		climxrrelem.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
36	6	fvexi 6518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑍 ∈ V)
38	9, 37	fexd 40840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ V)
39		eqidd 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
40	38, 39	clim 14718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
41	35, 40	mpbid 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
42	41	simpld 487	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
43	42	ad2antrr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐴 ∈ ℂ)
44	25, 43	subcld 10804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (-∞ − 𝐴) ∈ ℂ)
45	44	abscld 14663	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
46	45	adantlrr 709	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
47		climxrrelem.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
48	47	rpred 12254	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
49	48	ad2antrr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ∈ ℝ)
50	46, 49	ltnled 10593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ((abs‘(-∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴))))
51	34, 50	mpbid 224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴)))
52	28, 51	pm2.65da 805	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
53	52	3adant2 1112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
54	53	neqned 2976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ -∞)
55		simpll 755	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝜑)
56		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) = +∞)
57		simpl 475	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) ∈ ℂ)
58	56, 57	eqeltrrd 2869	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
59	58	adantll 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
60		climxrrelem.p	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
61	55, 59, 60	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
62	61	adantlrr 709	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
63		fvoveq1 7005	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = +∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
64	63	adantl 474	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
65		simpl 475	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
66	64, 65	eqbrtrrd 4958	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
67	66	adantll 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
68	67	adantlrl 708	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
69	42	ad2antrr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐴 ∈ ℂ)
70	59, 69	subcld 10804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (+∞ − 𝐴) ∈ ℂ)
71	70	abscld 14663	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
72	71	adantlrr 709	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
73	48	ad2antrr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ∈ ℝ)
74	72, 73	ltnled 10593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ((abs‘(+∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴))))
75	68, 74	mpbid 224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴)))
76	62, 75	pm2.65da 805	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
77	76	3adant2 1112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
78	77	neqned 2976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ +∞)
79	20, 54, 78	xrred 41097	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ)
80	14, 15, 18, 79	syl3anc 1352	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
81	13, 80	jca 504	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
82	5, 81	ralrimia 40858	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
83	9	ffund 6353	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
84		ffvresb 6717	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
85	83, 84	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
86	85	adantr 473	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
87	82, 86	mpbird 249	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
88		breq2 4938	. . . . . 6 ⊢ (𝑥 = 𝐷 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
89	88	anbi2d 620	. . . . 5 ⊢ (𝑥 = 𝐷 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
90	89	rexralbidv 3248	. . . 4 ⊢ (𝑥 = 𝐷 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
91	41	simprd 488	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
92	90, 91, 47	rspcdva 3543	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
93		climxrrelem.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
94	6	rexuz3 14575	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
95	93, 94	syl 17	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
96	92, 95	mpbird 249	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
97	87, 96	reximddv 3222	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3090 ∃wrex 3091 Vcvv 3417 class class class wbr 4934 dom cdm 5411 ↾ cres 5413 Fun wfun 6187 ⟶wf 6189 ‘cfv 6193 (class class class)co 6982 ℂcc 10339 ℝcr 10340 +∞cpnf 10477 -∞cmnf 10478 ℝ^*cxr 10479 < clt 10480 ≤ cle 10481 − cmin 10676 ℤcz 11799 ℤ_≥cuz 12064 ℝ⁺crp 12210 abscabs 14460 ⇝ cli 14708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-sup 8707 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-n0 11714 df-z 11800 df-uz 12065 df-rp 12211 df-seq 13191 df-exp 13251 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-clim 14712

This theorem is referenced by: climxrre 41497

W3C validator